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A scientist is studying the growth of two different populations of bacteria - Edexcel - A-Level Maths Pure - Question 9 - 2021 - Paper 1
Question 9
A scientist is studying the growth of two different populations of bacteria. The number of bacteria, N, in the first population is modelled by the equation N = Ae^{k... show full transcript
Worked Solution & Example Answer:A scientist is studying the growth of two different populations of bacteria - Edexcel - A-Level Maths Pure - Question 9 - 2021 - Paper 1
Step 1
find a complete equation for the model.
Answer
To find the complete equation, we start with the model given:
We know at the start of the study (t=0), the number of bacteria is 1000. Thus:
The equation now is:
Next, we use the information that the population doubles in 5 hours. Thus, at t=5:
Dividing both sides by 1000 gives:
Taking the natural logarithm:
So, we have:
k = rac{ ext{ln}(2)}{5}
Plugging this back into the equation:
N = 1000 e^{ rac{ ext{ln}(2)}{5}t}
Step 2
Hence find the rate of increase in the number of bacteria in this population exactly 8 hours from the start of the study.
Answer
To find the rate of increase, we differentiate the model:
N(t) = 1000 e^{ rac{ ext{ln}(2)}{5}t}
Thus,
rac{dN}{dt} = 1000 rac{ ext{ln}(2)}{5} e^{ rac{ ext{ln}(2)}{5}t}
Now, substituting t=8:
rac{dN}{dt} = 1000 rac{ ext{ln}(2)}{5} e^{ rac{ ext{ln}(2)}{5} imes 8}
Calculating:
rac{dN}{dt} = 1000 imes 0.138629436 imes e^{1.1456}. Evaluating gives approximately:
rac{dN}{dt} ext{ at } t=8 ext{ is approximately } 420 ext{ (to 2 significant figures)}
Step 3
find the value of T.
Answer
For the second population, we use:
We established earlier that:
k = rac{ ext{ln}(2)}{5}
Now we want to find T such that:
1000 e^{ rac{ ext{ln}(2)}{5}T} = 500 e^{ rac{ ext{ln}(2)}{5}T}
Cancelling out the exponential terms:
ightarrow e^{ rac{ ext{ln}(2)}{5}T}=2 $$ Taking natural logs: $$ rac{ ext{ln}(2)}{5}T = ext{ln}(2) $$ Thus: $$ T = 5 ext{ hours}$$